3.25.49 \(\int \frac {x^2}{\sqrt {a+b \sqrt {\frac {d}{x}}+\frac {c}{x}}} \, dx\)

Optimal. Leaf size=386 \[ \frac {b d^2 \left (156 a c-77 b^2 d\right ) \sqrt {a+b \sqrt {\frac {d}{x}}+\frac {c}{x}}}{160 a^4 \left (\frac {d}{x}\right )^{3/2}}-\frac {x^2 \left (100 a c-99 b^2 d\right ) \sqrt {a+b \sqrt {\frac {d}{x}}+\frac {c}{x}}}{240 a^3}-\frac {11 b d^3 \sqrt {a+b \sqrt {\frac {d}{x}}+\frac {c}{x}}}{30 a^2 \left (\frac {d}{x}\right )^{5/2}}-\frac {7 b d \left (528 a^2 c^2-680 a b^2 c d+165 b^4 d^2\right ) \sqrt {a+b \sqrt {\frac {d}{x}}+\frac {c}{x}}}{1280 a^6 \sqrt {\frac {d}{x}}}+\frac {x \left (400 a^2 c^2-1176 a b^2 c d+385 b^4 d^2\right ) \sqrt {a+b \sqrt {\frac {d}{x}}+\frac {c}{x}}}{640 a^5}-\frac {\left (320 a^3 c^3-1680 a^2 b^2 c^2 d+1260 a b^4 c d^2-231 b^6 d^3\right ) \tanh ^{-1}\left (\frac {2 a+b \sqrt {\frac {d}{x}}}{2 \sqrt {a} \sqrt {a+b \sqrt {\frac {d}{x}}+\frac {c}{x}}}\right )}{512 a^{13/2}}+\frac {x^3 \sqrt {a+b \sqrt {\frac {d}{x}}+\frac {c}{x}}}{3 a} \]

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Rubi [A]  time = 0.73, antiderivative size = 386, normalized size of antiderivative = 1.00, number of steps used = 10, number of rules used = 7, integrand size = 26, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.269, Rules used = {1970, 1357, 744, 834, 806, 724, 206} \begin {gather*} -\frac {7 b d \left (528 a^2 c^2-680 a b^2 c d+165 b^4 d^2\right ) \sqrt {a+b \sqrt {\frac {d}{x}}+\frac {c}{x}}}{1280 a^6 \sqrt {\frac {d}{x}}}+\frac {x \left (400 a^2 c^2-1176 a b^2 c d+385 b^4 d^2\right ) \sqrt {a+b \sqrt {\frac {d}{x}}+\frac {c}{x}}}{640 a^5}-\frac {\left (-1680 a^2 b^2 c^2 d+320 a^3 c^3+1260 a b^4 c d^2-231 b^6 d^3\right ) \tanh ^{-1}\left (\frac {2 a+b \sqrt {\frac {d}{x}}}{2 \sqrt {a} \sqrt {a+b \sqrt {\frac {d}{x}}+\frac {c}{x}}}\right )}{512 a^{13/2}}+\frac {b d^2 \left (156 a c-77 b^2 d\right ) \sqrt {a+b \sqrt {\frac {d}{x}}+\frac {c}{x}}}{160 a^4 \left (\frac {d}{x}\right )^{3/2}}-\frac {x^2 \left (100 a c-99 b^2 d\right ) \sqrt {a+b \sqrt {\frac {d}{x}}+\frac {c}{x}}}{240 a^3}-\frac {11 b d^3 \sqrt {a+b \sqrt {\frac {d}{x}}+\frac {c}{x}}}{30 a^2 \left (\frac {d}{x}\right )^{5/2}}+\frac {x^3 \sqrt {a+b \sqrt {\frac {d}{x}}+\frac {c}{x}}}{3 a} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[x^2/Sqrt[a + b*Sqrt[d/x] + c/x],x]

[Out]

(-11*b*d^3*Sqrt[a + b*Sqrt[d/x] + c/x])/(30*a^2*(d/x)^(5/2)) + (b*d^2*(156*a*c - 77*b^2*d)*Sqrt[a + b*Sqrt[d/x
] + c/x])/(160*a^4*(d/x)^(3/2)) - (7*b*d*(528*a^2*c^2 - 680*a*b^2*c*d + 165*b^4*d^2)*Sqrt[a + b*Sqrt[d/x] + c/
x])/(1280*a^6*Sqrt[d/x]) + ((400*a^2*c^2 - 1176*a*b^2*c*d + 385*b^4*d^2)*Sqrt[a + b*Sqrt[d/x] + c/x]*x)/(640*a
^5) - ((100*a*c - 99*b^2*d)*Sqrt[a + b*Sqrt[d/x] + c/x]*x^2)/(240*a^3) + (Sqrt[a + b*Sqrt[d/x] + c/x]*x^3)/(3*
a) - ((320*a^3*c^3 - 1680*a^2*b^2*c^2*d + 1260*a*b^4*c*d^2 - 231*b^6*d^3)*ArcTanh[(2*a + b*Sqrt[d/x])/(2*Sqrt[
a]*Sqrt[a + b*Sqrt[d/x] + c/x])])/(512*a^(13/2))

Rule 206

Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(1*ArcTanh[(Rt[-b, 2]*x)/Rt[a, 2]])/(Rt[a, 2]*Rt[-b, 2]), x]
/; FreeQ[{a, b}, x] && NegQ[a/b] && (GtQ[a, 0] || LtQ[b, 0])

Rule 724

Int[1/(((d_.) + (e_.)*(x_))*Sqrt[(a_.) + (b_.)*(x_) + (c_.)*(x_)^2]), x_Symbol] :> Dist[-2, Subst[Int[1/(4*c*d
^2 - 4*b*d*e + 4*a*e^2 - x^2), x], x, (2*a*e - b*d - (2*c*d - b*e)*x)/Sqrt[a + b*x + c*x^2]], x] /; FreeQ[{a,
b, c, d, e}, x] && NeQ[b^2 - 4*a*c, 0] && NeQ[2*c*d - b*e, 0]

Rule 744

Int[((d_.) + (e_.)*(x_))^(m_)*((a_.) + (b_.)*(x_) + (c_.)*(x_)^2)^(p_), x_Symbol] :> Simp[(e*(d + e*x)^(m + 1)
*(a + b*x + c*x^2)^(p + 1))/((m + 1)*(c*d^2 - b*d*e + a*e^2)), x] + Dist[1/((m + 1)*(c*d^2 - b*d*e + a*e^2)),
Int[(d + e*x)^(m + 1)*Simp[c*d*(m + 1) - b*e*(m + p + 2) - c*e*(m + 2*p + 3)*x, x]*(a + b*x + c*x^2)^p, x], x]
/; FreeQ[{a, b, c, d, e, m, p}, x] && NeQ[b^2 - 4*a*c, 0] && NeQ[c*d^2 - b*d*e + a*e^2, 0] && NeQ[2*c*d - b*e
, 0] && NeQ[m, -1] && ((LtQ[m, -1] && IntQuadraticQ[a, b, c, d, e, m, p, x]) || (SumSimplerQ[m, 1] && IntegerQ
[p]) || ILtQ[Simplify[m + 2*p + 3], 0])

Rule 806

Int[((d_.) + (e_.)*(x_))^(m_)*((f_.) + (g_.)*(x_))*((a_.) + (b_.)*(x_) + (c_.)*(x_)^2)^(p_.), x_Symbol] :> -Si
mp[((e*f - d*g)*(d + e*x)^(m + 1)*(a + b*x + c*x^2)^(p + 1))/(2*(p + 1)*(c*d^2 - b*d*e + a*e^2)), x] - Dist[(b
*(e*f + d*g) - 2*(c*d*f + a*e*g))/(2*(c*d^2 - b*d*e + a*e^2)), Int[(d + e*x)^(m + 1)*(a + b*x + c*x^2)^p, x],
x] /; FreeQ[{a, b, c, d, e, f, g, m, p}, x] && NeQ[b^2 - 4*a*c, 0] && NeQ[c*d^2 - b*d*e + a*e^2, 0] && EqQ[Sim
plify[m + 2*p + 3], 0]

Rule 834

Int[((d_.) + (e_.)*(x_))^(m_)*((f_.) + (g_.)*(x_))*((a_.) + (b_.)*(x_) + (c_.)*(x_)^2)^(p_.), x_Symbol] :> Sim
p[((e*f - d*g)*(d + e*x)^(m + 1)*(a + b*x + c*x^2)^(p + 1))/((m + 1)*(c*d^2 - b*d*e + a*e^2)), x] + Dist[1/((m
+ 1)*(c*d^2 - b*d*e + a*e^2)), Int[(d + e*x)^(m + 1)*(a + b*x + c*x^2)^p*Simp[(c*d*f - f*b*e + a*e*g)*(m + 1)
+ b*(d*g - e*f)*(p + 1) - c*(e*f - d*g)*(m + 2*p + 3)*x, x], x], x] /; FreeQ[{a, b, c, d, e, f, g, p}, x] &&
NeQ[b^2 - 4*a*c, 0] && NeQ[c*d^2 - b*d*e + a*e^2, 0] && LtQ[m, -1] && (IntegerQ[m] || IntegerQ[p] || IntegersQ
[2*m, 2*p])

Rule 1357

Int[(x_)^(m_.)*((a_) + (c_.)*(x_)^(n2_.) + (b_.)*(x_)^(n_))^(p_.), x_Symbol] :> Dist[1/n, Subst[Int[x^(Simplif
y[(m + 1)/n] - 1)*(a + b*x + c*x^2)^p, x], x, x^n], x] /; FreeQ[{a, b, c, m, n, p}, x] && EqQ[n2, 2*n] && NeQ[
b^2 - 4*a*c, 0] && IntegerQ[Simplify[(m + 1)/n]]

Rule 1970

Int[(x_)^(m_.)*((a_) + (b_.)*((d_.)/(x_))^(n_) + (c_.)*(x_)^(n2_.))^(p_.), x_Symbol] :> -Dist[d^(m + 1), Subst
[Int[(a + b*x^n + (c*x^(2*n))/d^(2*n))^p/x^(m + 2), x], x, d/x], x] /; FreeQ[{a, b, c, d, n, p}, x] && EqQ[n2,
-2*n] && IntegerQ[2*n] && IntegerQ[m]

Rubi steps

\begin {align*} \int \frac {x^2}{\sqrt {a+b \sqrt {\frac {d}{x}}+\frac {c}{x}}} \, dx &=-\left (d^3 \operatorname {Subst}\left (\int \frac {1}{x^4 \sqrt {a+b \sqrt {x}+\frac {c x}{d}}} \, dx,x,\frac {d}{x}\right )\right )\\ &=-\left (\left (2 d^3\right ) \operatorname {Subst}\left (\int \frac {1}{x^7 \sqrt {a+b x+\frac {c x^2}{d}}} \, dx,x,\sqrt {\frac {d}{x}}\right )\right )\\ &=\frac {\sqrt {a+b \sqrt {\frac {d}{x}}+\frac {c}{x}} x^3}{3 a}+\frac {d^3 \operatorname {Subst}\left (\int \frac {\frac {11 b}{2}+\frac {5 c x}{d}}{x^6 \sqrt {a+b x+\frac {c x^2}{d}}} \, dx,x,\sqrt {\frac {d}{x}}\right )}{3 a}\\ &=-\frac {11 b d^3 \sqrt {a+b \sqrt {\frac {d}{x}}+\frac {c}{x}}}{30 a^2 \left (\frac {d}{x}\right )^{5/2}}+\frac {\sqrt {a+b \sqrt {\frac {d}{x}}+\frac {c}{x}} x^3}{3 a}-\frac {d^3 \operatorname {Subst}\left (\int \frac {\frac {1}{4} \left (99 b^2-\frac {100 a c}{d}\right )+\frac {22 b c x}{d}}{x^5 \sqrt {a+b x+\frac {c x^2}{d}}} \, dx,x,\sqrt {\frac {d}{x}}\right )}{15 a^2}\\ &=-\frac {11 b d^3 \sqrt {a+b \sqrt {\frac {d}{x}}+\frac {c}{x}}}{30 a^2 \left (\frac {d}{x}\right )^{5/2}}-\frac {\left (100 a c-99 b^2 d\right ) \sqrt {a+b \sqrt {\frac {d}{x}}+\frac {c}{x}} x^2}{240 a^3}+\frac {\sqrt {a+b \sqrt {\frac {d}{x}}+\frac {c}{x}} x^3}{3 a}+\frac {d^3 \operatorname {Subst}\left (\int \frac {-\frac {9 b \left (156 a c-77 b^2 d\right )}{8 d}-\frac {3 c \left (100 a c-99 b^2 d\right ) x}{4 d^2}}{x^4 \sqrt {a+b x+\frac {c x^2}{d}}} \, dx,x,\sqrt {\frac {d}{x}}\right )}{60 a^3}\\ &=-\frac {11 b d^3 \sqrt {a+b \sqrt {\frac {d}{x}}+\frac {c}{x}}}{30 a^2 \left (\frac {d}{x}\right )^{5/2}}+\frac {b d^2 \left (156 a c-77 b^2 d\right ) \sqrt {a+b \sqrt {\frac {d}{x}}+\frac {c}{x}}}{160 a^4 \left (\frac {d}{x}\right )^{3/2}}-\frac {\left (100 a c-99 b^2 d\right ) \sqrt {a+b \sqrt {\frac {d}{x}}+\frac {c}{x}} x^2}{240 a^3}+\frac {\sqrt {a+b \sqrt {\frac {d}{x}}+\frac {c}{x}} x^3}{3 a}-\frac {d^3 \operatorname {Subst}\left (\int \frac {\frac {9 \left (400 a^2 c^2-1176 a b^2 c d+385 b^4 d^2\right )}{16 d^2}-\frac {9 b c \left (156 a c-77 b^2 d\right ) x}{4 d^2}}{x^3 \sqrt {a+b x+\frac {c x^2}{d}}} \, dx,x,\sqrt {\frac {d}{x}}\right )}{180 a^4}\\ &=-\frac {11 b d^3 \sqrt {a+b \sqrt {\frac {d}{x}}+\frac {c}{x}}}{30 a^2 \left (\frac {d}{x}\right )^{5/2}}+\frac {b d^2 \left (156 a c-77 b^2 d\right ) \sqrt {a+b \sqrt {\frac {d}{x}}+\frac {c}{x}}}{160 a^4 \left (\frac {d}{x}\right )^{3/2}}+\frac {\left (400 a^2 c^2-1176 a b^2 c d+385 b^4 d^2\right ) \sqrt {a+b \sqrt {\frac {d}{x}}+\frac {c}{x}} x}{640 a^5}-\frac {\left (100 a c-99 b^2 d\right ) \sqrt {a+b \sqrt {\frac {d}{x}}+\frac {c}{x}} x^2}{240 a^3}+\frac {\sqrt {a+b \sqrt {\frac {d}{x}}+\frac {c}{x}} x^3}{3 a}+\frac {d^3 \operatorname {Subst}\left (\int \frac {\frac {63 b \left (528 a^2 c^2-680 a b^2 c d+165 b^4 d^2\right )}{32 d^2}+\frac {9 c \left (400 a^2 c^2-1176 a b^2 c d+385 b^4 d^2\right ) x}{16 d^3}}{x^2 \sqrt {a+b x+\frac {c x^2}{d}}} \, dx,x,\sqrt {\frac {d}{x}}\right )}{360 a^5}\\ &=-\frac {11 b d^3 \sqrt {a+b \sqrt {\frac {d}{x}}+\frac {c}{x}}}{30 a^2 \left (\frac {d}{x}\right )^{5/2}}+\frac {b d^2 \left (156 a c-77 b^2 d\right ) \sqrt {a+b \sqrt {\frac {d}{x}}+\frac {c}{x}}}{160 a^4 \left (\frac {d}{x}\right )^{3/2}}-\frac {7 b d \left (528 a^2 c^2-680 a b^2 c d+165 b^4 d^2\right ) \sqrt {a+b \sqrt {\frac {d}{x}}+\frac {c}{x}}}{1280 a^6 \sqrt {\frac {d}{x}}}+\frac {\left (400 a^2 c^2-1176 a b^2 c d+385 b^4 d^2\right ) \sqrt {a+b \sqrt {\frac {d}{x}}+\frac {c}{x}} x}{640 a^5}-\frac {\left (100 a c-99 b^2 d\right ) \sqrt {a+b \sqrt {\frac {d}{x}}+\frac {c}{x}} x^2}{240 a^3}+\frac {\sqrt {a+b \sqrt {\frac {d}{x}}+\frac {c}{x}} x^3}{3 a}+\frac {\left (320 a^3 c^3-1680 a^2 b^2 c^2 d+1260 a b^4 c d^2-231 b^6 d^3\right ) \operatorname {Subst}\left (\int \frac {1}{x \sqrt {a+b x+\frac {c x^2}{d}}} \, dx,x,\sqrt {\frac {d}{x}}\right )}{512 a^6}\\ &=-\frac {11 b d^3 \sqrt {a+b \sqrt {\frac {d}{x}}+\frac {c}{x}}}{30 a^2 \left (\frac {d}{x}\right )^{5/2}}+\frac {b d^2 \left (156 a c-77 b^2 d\right ) \sqrt {a+b \sqrt {\frac {d}{x}}+\frac {c}{x}}}{160 a^4 \left (\frac {d}{x}\right )^{3/2}}-\frac {7 b d \left (528 a^2 c^2-680 a b^2 c d+165 b^4 d^2\right ) \sqrt {a+b \sqrt {\frac {d}{x}}+\frac {c}{x}}}{1280 a^6 \sqrt {\frac {d}{x}}}+\frac {\left (400 a^2 c^2-1176 a b^2 c d+385 b^4 d^2\right ) \sqrt {a+b \sqrt {\frac {d}{x}}+\frac {c}{x}} x}{640 a^5}-\frac {\left (100 a c-99 b^2 d\right ) \sqrt {a+b \sqrt {\frac {d}{x}}+\frac {c}{x}} x^2}{240 a^3}+\frac {\sqrt {a+b \sqrt {\frac {d}{x}}+\frac {c}{x}} x^3}{3 a}-\frac {\left (320 a^3 c^3-1680 a^2 b^2 c^2 d+1260 a b^4 c d^2-231 b^6 d^3\right ) \operatorname {Subst}\left (\int \frac {1}{4 a-x^2} \, dx,x,\frac {2 a+b \sqrt {\frac {d}{x}}}{\sqrt {a+b \sqrt {\frac {d}{x}}+\frac {c}{x}}}\right )}{256 a^6}\\ &=-\frac {11 b d^3 \sqrt {a+b \sqrt {\frac {d}{x}}+\frac {c}{x}}}{30 a^2 \left (\frac {d}{x}\right )^{5/2}}+\frac {b d^2 \left (156 a c-77 b^2 d\right ) \sqrt {a+b \sqrt {\frac {d}{x}}+\frac {c}{x}}}{160 a^4 \left (\frac {d}{x}\right )^{3/2}}-\frac {7 b d \left (528 a^2 c^2-680 a b^2 c d+165 b^4 d^2\right ) \sqrt {a+b \sqrt {\frac {d}{x}}+\frac {c}{x}}}{1280 a^6 \sqrt {\frac {d}{x}}}+\frac {\left (400 a^2 c^2-1176 a b^2 c d+385 b^4 d^2\right ) \sqrt {a+b \sqrt {\frac {d}{x}}+\frac {c}{x}} x}{640 a^5}-\frac {\left (100 a c-99 b^2 d\right ) \sqrt {a+b \sqrt {\frac {d}{x}}+\frac {c}{x}} x^2}{240 a^3}+\frac {\sqrt {a+b \sqrt {\frac {d}{x}}+\frac {c}{x}} x^3}{3 a}-\frac {\left (320 a^3 c^3-1680 a^2 b^2 c^2 d+1260 a b^4 c d^2-231 b^6 d^3\right ) \tanh ^{-1}\left (\frac {2 a+b \sqrt {\frac {d}{x}}}{2 \sqrt {a} \sqrt {a+b \sqrt {\frac {d}{x}}+\frac {c}{x}}}\right )}{512 a^{13/2}}\\ \end {align*}

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Mathematica [F]  time = 0.41, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {x^2}{\sqrt {a+b \sqrt {\frac {d}{x}}+\frac {c}{x}}} \, dx \end {gather*}

Verification is not applicable to the result.

[In]

Integrate[x^2/Sqrt[a + b*Sqrt[d/x] + c/x],x]

[Out]

Integrate[x^2/Sqrt[a + b*Sqrt[d/x] + c/x], x]

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IntegrateAlgebraic [A]  time = 1.82, size = 323, normalized size = 0.84 \begin {gather*} \frac {\left (320 a^3 c^3-1680 a^2 b^2 c^2 d+1260 a b^4 c d^2-231 b^6 d^3\right ) \tanh ^{-1}\left (\frac {\sqrt {\frac {c}{d}} \sqrt {\frac {d}{x}}-\sqrt {a+b \sqrt {\frac {d}{x}}+\frac {c}{x}}}{\sqrt {a}}\right )}{256 a^{13/2}}+\frac {x^3 \sqrt {a+b \sqrt {\frac {d}{x}}+\frac {c}{x}} \left (1280 a^5 d^3-1408 a^4 b d^3 \sqrt {\frac {d}{x}}-\frac {1600 a^4 c d^3}{x}+\frac {1584 a^3 b^2 d^4}{x}+3744 a^3 b c d^2 \left (\frac {d}{x}\right )^{3/2}+\frac {2400 a^3 c^2 d^3}{x^2}-1848 a^2 b^3 d^3 \left (\frac {d}{x}\right )^{3/2}-\frac {7056 a^2 b^2 c d^4}{x^2}-11088 a^2 b c^2 d \left (\frac {d}{x}\right )^{5/2}+\frac {2310 a b^4 d^5}{x^2}+14280 a b^3 c d^2 \left (\frac {d}{x}\right )^{5/2}-3465 b^5 d^3 \left (\frac {d}{x}\right )^{5/2}\right )}{3840 a^6 d^3} \end {gather*}

Antiderivative was successfully verified.

[In]

IntegrateAlgebraic[x^2/Sqrt[a + b*Sqrt[d/x] + c/x],x]

[Out]

(Sqrt[a + b*Sqrt[d/x] + c/x]*(1280*a^5*d^3 - 1408*a^4*b*d^3*Sqrt[d/x] + 3744*a^3*b*c*d^2*(d/x)^(3/2) - 1848*a^
2*b^3*d^3*(d/x)^(3/2) - 11088*a^2*b*c^2*d*(d/x)^(5/2) + 14280*a*b^3*c*d^2*(d/x)^(5/2) - 3465*b^5*d^3*(d/x)^(5/
2) + (2400*a^3*c^2*d^3)/x^2 - (7056*a^2*b^2*c*d^4)/x^2 + (2310*a*b^4*d^5)/x^2 - (1600*a^4*c*d^3)/x + (1584*a^3
*b^2*d^4)/x)*x^3)/(3840*a^6*d^3) + ((320*a^3*c^3 - 1680*a^2*b^2*c^2*d + 1260*a*b^4*c*d^2 - 231*b^6*d^3)*ArcTan
h[(-Sqrt[a + b*Sqrt[d/x] + c/x] + Sqrt[c/d]*Sqrt[d/x])/Sqrt[a]])/(256*a^(13/2))

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fricas [F(-1)]  time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Timed out} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^2/(a+c/x+b*(d/x)^(1/2))^(1/2),x, algorithm="fricas")

[Out]

Timed out

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giac [F(-2)]  time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Exception raised: TypeError} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^2/(a+c/x+b*(d/x)^(1/2))^(1/2),x, algorithm="giac")

[Out]

Exception raised: TypeError >> An error occurred running a Giac command:INPUT:sage2:=int(sage0,x):;OUTPUT:sym2
poly/r2sym(const gen & e,const index_m & i,const vecteur & l) Error: Bad Argument Valuesym2poly/r2sym(const ge
n & e,const index_m & i,const vecteur & l) Error: Bad Argument ValueWarning, integration of abs or sign assume
s constant sign by intervals (correct if the argument is real):Check [abs(t_nostep)]Warning, integration of ab
s or sign assumes constant sign by intervals (correct if the argument is real):Check [abs(t_nostep)]Evaluation
time: 0.53Done

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maple [A]  time = 0.16, size = 655, normalized size = 1.70 \begin {gather*} \frac {\sqrt {\frac {a x +\sqrt {\frac {d}{x}}\, b x +c}{x}}\, \left (3465 a \,b^{6} d^{3} \ln \left (\frac {2 a \sqrt {x}+\sqrt {\frac {d}{x}}\, b \sqrt {x}+2 \sqrt {a x +\sqrt {\frac {d}{x}}\, b x +c}\, \sqrt {a}}{2 \sqrt {a}}\right )-18900 a^{2} b^{4} c \,d^{2} \ln \left (\frac {2 a \sqrt {x}+\sqrt {\frac {d}{x}}\, b \sqrt {x}+2 \sqrt {a x +\sqrt {\frac {d}{x}}\, b x +c}\, \sqrt {a}}{2 \sqrt {a}}\right )+2560 \sqrt {a x +\sqrt {\frac {d}{x}}\, b x +c}\, a^{\frac {13}{2}} x^{\frac {5}{2}}-2816 \sqrt {a x +\sqrt {\frac {d}{x}}\, b x +c}\, \sqrt {\frac {d}{x}}\, a^{\frac {11}{2}} b \,x^{\frac {5}{2}}+3168 \sqrt {a x +\sqrt {\frac {d}{x}}\, b x +c}\, a^{\frac {9}{2}} b^{2} d \,x^{\frac {3}{2}}-3696 \sqrt {a x +\sqrt {\frac {d}{x}}\, b x +c}\, \left (\frac {d}{x}\right )^{\frac {3}{2}} a^{\frac {7}{2}} b^{3} x^{\frac {5}{2}}+25200 a^{3} b^{2} c^{2} d \ln \left (\frac {2 a \sqrt {x}+\sqrt {\frac {d}{x}}\, b \sqrt {x}+2 \sqrt {a x +\sqrt {\frac {d}{x}}\, b x +c}\, \sqrt {a}}{2 \sqrt {a}}\right )+4620 \sqrt {a x +\sqrt {\frac {d}{x}}\, b x +c}\, a^{\frac {5}{2}} b^{4} d^{2} \sqrt {x}-6930 \sqrt {a x +\sqrt {\frac {d}{x}}\, b x +c}\, \left (\frac {d}{x}\right )^{\frac {5}{2}} a^{\frac {3}{2}} b^{5} x^{\frac {5}{2}}-3200 \sqrt {a x +\sqrt {\frac {d}{x}}\, b x +c}\, a^{\frac {11}{2}} c \,x^{\frac {3}{2}}+7488 \sqrt {a x +\sqrt {\frac {d}{x}}\, b x +c}\, \sqrt {\frac {d}{x}}\, a^{\frac {9}{2}} b c \,x^{\frac {3}{2}}-4800 a^{4} c^{3} \ln \left (\frac {2 a \sqrt {x}+\sqrt {\frac {d}{x}}\, b \sqrt {x}+2 \sqrt {a x +\sqrt {\frac {d}{x}}\, b x +c}\, \sqrt {a}}{2 \sqrt {a}}\right )-14112 \sqrt {a x +\sqrt {\frac {d}{x}}\, b x +c}\, a^{\frac {7}{2}} b^{2} c d \sqrt {x}+28560 \sqrt {a x +\sqrt {\frac {d}{x}}\, b x +c}\, \left (\frac {d}{x}\right )^{\frac {3}{2}} a^{\frac {5}{2}} b^{3} c \,x^{\frac {3}{2}}+4800 \sqrt {a x +\sqrt {\frac {d}{x}}\, b x +c}\, a^{\frac {9}{2}} c^{2} \sqrt {x}-22176 \sqrt {a x +\sqrt {\frac {d}{x}}\, b x +c}\, \sqrt {\frac {d}{x}}\, a^{\frac {7}{2}} b \,c^{2} \sqrt {x}\right ) \sqrt {x}}{7680 \sqrt {a x +\sqrt {\frac {d}{x}}\, b x +c}\, a^{\frac {15}{2}}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(x^2/(a+(d/x)^(1/2)*b+c/x)^(1/2),x)

[Out]

1/7680*((a*x+(d/x)^(1/2)*b*x+c)/x)^(1/2)*x^(1/2)*(2560*x^(5/2)*(a*x+(d/x)^(1/2)*b*x+c)^(1/2)*a^(13/2)-2816*(a*
x+(d/x)^(1/2)*b*x+c)^(1/2)*a^(11/2)*(d/x)^(1/2)*x^(5/2)*b-3696*(a*x+(d/x)^(1/2)*b*x+c)^(1/2)*a^(7/2)*(d/x)^(3/
2)*x^(5/2)*b^3-6930*(a*x+(d/x)^(1/2)*b*x+c)^(1/2)*(d/x)^(5/2)*a^(3/2)*b^5*x^(5/2)-3200*(a*x+(d/x)^(1/2)*b*x+c)
^(1/2)*a^(11/2)*x^(3/2)*c+7488*(a*x+(d/x)^(1/2)*b*x+c)^(1/2)*a^(9/2)*(d/x)^(1/2)*x^(3/2)*b*c+3168*(a*x+(d/x)^(
1/2)*b*x+c)^(1/2)*d*a^(9/2)*x^(3/2)*b^2+28560*(a*x+(d/x)^(1/2)*b*x+c)^(1/2)*(d/x)^(3/2)*a^(5/2)*b^3*c*x^(3/2)+
4800*(a*x+(d/x)^(1/2)*b*x+c)^(1/2)*a^(9/2)*c^2*x^(1/2)-22176*(a*x+(d/x)^(1/2)*b*x+c)^(1/2)*(d/x)^(1/2)*a^(7/2)
*b*c^2*x^(1/2)-14112*(a*x+(d/x)^(1/2)*b*x+c)^(1/2)*a^(7/2)*b^2*c*d*x^(1/2)+4620*(a*x+(d/x)^(1/2)*b*x+c)^(1/2)*
a^(5/2)*b^4*d^2*x^(1/2)+3465*a*b^6*d^3*ln(1/2*(2*a*x^(1/2)+(d/x)^(1/2)*b*x^(1/2)+2*(a*x+(d/x)^(1/2)*b*x+c)^(1/
2)*a^(1/2))/a^(1/2))-18900*a^2*b^4*c*d^2*ln(1/2*(2*a*x^(1/2)+(d/x)^(1/2)*b*x^(1/2)+2*(a*x+(d/x)^(1/2)*b*x+c)^(
1/2)*a^(1/2))/a^(1/2))+25200*a^3*b^2*c^2*d*ln(1/2*(2*a*x^(1/2)+(d/x)^(1/2)*b*x^(1/2)+2*(a*x+(d/x)^(1/2)*b*x+c)
^(1/2)*a^(1/2))/a^(1/2))-4800*a^4*c^3*ln(1/2*(2*a*x^(1/2)+(d/x)^(1/2)*b*x^(1/2)+2*(a*x+(d/x)^(1/2)*b*x+c)^(1/2
)*a^(1/2))/a^(1/2)))/(a*x+(d/x)^(1/2)*b*x+c)^(1/2)/a^(15/2)

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maxima [F]  time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {x^{2}}{\sqrt {b \sqrt {\frac {d}{x}} + a + \frac {c}{x}}}\,{d x} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^2/(a+c/x+b*(d/x)^(1/2))^(1/2),x, algorithm="maxima")

[Out]

integrate(x^2/sqrt(b*sqrt(d/x) + a + c/x), x)

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mupad [F]  time = 0.00, size = -1, normalized size = -0.00 \begin {gather*} \int \frac {x^2}{\sqrt {a+\frac {c}{x}+b\,\sqrt {\frac {d}{x}}}} \,d x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(x^2/(a + c/x + b*(d/x)^(1/2))^(1/2),x)

[Out]

int(x^2/(a + c/x + b*(d/x)^(1/2))^(1/2), x)

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sympy [F]  time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {x^{2}}{\sqrt {a + b \sqrt {\frac {d}{x}} + \frac {c}{x}}}\, dx \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x**2/(a+c/x+b*(d/x)**(1/2))**(1/2),x)

[Out]

Integral(x**2/sqrt(a + b*sqrt(d/x) + c/x), x)

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